A survey of Internet users reported that 15% downloaded music onto their compute

Question

A survey of Internet users reported that 15% downloaded music onto their computers. The filing of lawsuits by the recording industry may be a reason why this percent has decreased from the estimate of 25% from a survey taken two years before. Suppose we are not exactly sure about the sizes of the samples. Perform the calculations for the significance tests and 95% confidence intervals under each of the following assumptions. (Use previous recent. Round your test statistics to two decimal places and your confidence intervals to four decimal places.)

(i) Both sample sizes are 1000.

z=

95% C.I.=

(ii) Both sample sizes are 1600.

z=

95% C.I.=

(iii) The sample size for the survey reporting 25% is 1000 and the sample size for the survey reporting 15% is 1600.

z=

95% C.I.=

Summarize the effects of the sample sizes on the results.

We see in (i) and (ii) that smaller samples result in larger z (stronger evidence) and narrower intervals, while larger samples have the reverse effect. The results of (iii) show that the effect of varying unequal sample sizes is more complicated.

We see in (i) and (ii) that smaller samples result in smaller z (weaker evidence) and wider intervals, while larger samples have the reverse effect. The results of (iii) show that the effect of varying unequal sample sizes is more complicated.

We see in (i) and (ii) that smaller samples result in larger z (weaker evidence) and smaller intervals, while larger samples have the reverse effect. The results of (iii) show that the effect of varying unequal sample sizes is more complicated.

We see in (i) and (ii) that smaller samples result in smaller z (stronger evidence) and wider intervals, while larger samples have the reverse effect. The results of (iii) show that the effect of varying unequal sample sizes is more complicated.

We see in (i) and (ii) that smaller samples result in smaller z (weaker evidence) and narrower intervals, while larger samples have the reverse effect. The results of (iii) show that the effect of varying unequal sample sizes is more complicated.

Explanation / Answer

i.

Formulating the hypotheses          
Ho: p1 - p2   =   0  
Ha: p1 - p2   =/=   0  
Here, we see that pdo =    0   , the hypothesized population proportion difference.  
          
Getting p1^ and p2^,          
          
p1^ = x1/n1 =    0.15      
p2 = x2/n2 =    0.25      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.017748239      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    -5.634361698   [ANSWER, Z]  

**********************************
          
For the   95%   confidence level, then  
          
alpha/2 = (1 - confidence level)/2 =    0.025      
z(alpha/2) =    1.959963985      
Margin of error = z(alpha/2)*sd =    0.03478591      
lower bound = p1^ - p2^ - z(alpha/2) * sd =    -0.13478591      
upper bound = p1^ - p2^ + z(alpha/2) * sd =    -0.06521409      
          
Thus, the confidence interval is          
          
(   -0.13478591   ,   -0.06521409 ) [ANSWER, CI]

*****************************************************

ii.

Formulating the hypotheses          
Ho: p1 - p2   =   0  
Ha: p1 - p2   =/=   0  
Here, we see that pdo =    0   , the hypothesized population proportion difference.  
          
Getting p1^ and p2^,          
          
p1^ = x1/n1 =    0.15      
p2 = x2/n2 =    0.25      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.014031215      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    -7.126966451   [ANSWER, Z]

**********************  
          
For the   95%   confidence level, then  
          
alpha/2 = (1 - confidence level)/2 =    0.025      
z(alpha/2) =    1.959963985      
Margin of error = z(alpha/2)*sd =    0.027500676      
lower bound = p1^ - p2^ - z(alpha/2) * sd =    -0.127500676      
upper bound = p1^ - p2^ + z(alpha/2) * sd =    -0.072499324      
          
Thus, the confidence interval is          
          
(   -0.127500676   ,   -0.072499324 ) [ANSWER, CI]

****************************************************

*******************************************

Hi! Please submit the next part as a separate question. That way we can continue helping you! Please indicate which parts are not yet solved when you submit. Thanks!

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.